Research

I am primarily interested in
algorithmic research. My research interests expand to
design and analysis of algorithms, approximation algorithms
in both sequential and distributed models of computation,
graph theory, combinatorial optimization, and computational
geometry. The focus of my research these days is algorithmic
problems on reasonable graph theoretic/geometric models for wireless
networks motivated by real problems in such networks.
Here's
a research statement detailing my research interests and work.
The following is a list of the papers that I have co-authored:

Journal:

Shifting Strategy for Graphs Without Geometry.

To appear in the Special Issue of Journal of
Combinatorial Optimization
dedicated to ISAAC 2009.

Clustering to Minimize the Sum of Radii.

With Matt Gibson, Gaurav Kanade, Erik Krohn, and
Kasturi Varadarajan.
To appear in the Special Issue of Algorithmica
dedicated to SWAT 2008.

Topology Control and Geographic Routing in Realistic
Wireless Networks.

Conference:

Consider the intersecction graph defined by a collection of n disks
in the plane. Our goal is to find a dominating of minimum cost. The
problem has been extensively studied on various graph classes, including
subclasses of disk graphs, yet, for disk graphs, nothing better than
an O(log n) approximation was known. We improve the status quo in
two ways: for the unweighted case, we show that a simple local
search yields a PTAS; for the weighted case, we give a
2O(log*n) approximation to an optimum solution.
The first approach follows the framework of Mustafa-Ray [SoCG09] and
Chan-Har-Peled [SoCG09], while the second follows the framework of
Varadarajan [STOC10].

We consider the problem of partitioning the vertex set of an
input unit disk graph (UDG) into the fewest number of cliques;
the problem is NP-hard. Various O(1) approximations are easily
obtainable. We give a first PTAS for this problem. In fact, we give a
(weakly) robust algorithm that takes as input a graph with associated
edge-lengths (instead of coordinates for the vertices). If the input
graph has a realization in the Euclidean plane, then our algorithm
is a (1+ε)-approximation. If the input has no feasible UDG embedding,
then the
algorithm either terminates producing a certificate proving that the
graph is not a UDG, or produces a clique partition without any
guarantees on the size of the partition.
We also consider a weighted version of the problem where even the
edge-lengths aren't given -- the graph is expressed in general form. The
vertices have weights. The weight of a clique is the weight of the
heaviest vertex in the clique, and the goal is to produce a clique
partition with the lightest weight. This generalizes the classical
problem. We give a (weakly) robust (2+ε)-approximation for this
problem. This improves on the best known 8-approximation for the
unweighted version of the problem.

We give a simple framework which is an alternative to the
celebrated shifting strategy of Hochbaum and Maass
[J. ACM, 1985]. Our framework does not require the input graph
to have a geometric realization -- it only requires that the input
graph satisfy some weak property referred to as growth
boundedness. As an application of the framework
we show how to obtain PTAS for maximum (weighted) independent
set problem on this graph class; the problem is NP-hard.
Via a more sophisticated application of our framework, we also show
how to obtain a PTAS for the maximum (weighted) independent set
for intersection graphs of (low-dimensional) fat objects that
are expressed without geometry. Erlebach et al. [SIAM J. Comput.,
2005]
and Chan [J. Algorithms, 2003] gave a PTAS
for this problem for intersection graphs
of fat objects. Their scheme required a geometric representation of
the input. Our result gives a positive
answer to a question of Erlebach et al.
who asked if a PTAS for this problem can be obtained without access
to geometry.

Given an n-point metric (P,d) and an integer k > 0, we consider
the problem of covering P by k balls so as to minimize the sum
of the radii of the balls. We present a randomized algorithm that
runs in nO(log n log Δ) time and returns with
high probability the optimal solution. Here, Δ is the ratio
between the maximum and minimum interpoint distances in the metric
space. We also show that the problem
is NP-hard, even in metrics induced by weighted planar graphs and
in metrics of constant doubling dimension.

Given a set of n input points in the Euclidean plane and
an integer k, the goal is to place at most k disks at
the input points so as to cover all of them. Define
the cost of a disk as its radius and a solution as the
sum of costs of the disks. The goal is to find a minimum
cost cover. We give an exact algorithm for this problem
that runs in poly-time if two sums of square roots of
rational number can be compared in poly-time.

Given a unit disk graph (UDG) in general form,
we consider the problem of devising a "good quality"
embedding in the Euclidean plane for it. "Quality" of an
embedding has been defined as the ratio of the length of
the longest edge to the length of the shortest non-edge.
We give a combinatorial algorithm that gives a
O(log2.5 n)
O(log3 n) quality embedding
of an input UDG specified in general form. We also show
how to obtain a quality-2 embedding of the UDG if we are
allowed to embed the graph in O(1)-dimensional
Euclidean space, instead of strictly a Euclidean plane.
In the preliminary version which appeared in ESA-07,
there is a technical flaw in obtaining a
(k,O(log0.5n)) volume respecting embedding of the
cluster graph. Also, our result only
applies to UDGs and not UBGs.

Given a alpha-quasi-unit disk graph (qUDG)
(alpha >= 1/sqrt(2)) we consider the problem of robust and
memoryless geographic routing on such graphs. We compute,
in O(1) rounds of distributed computation, a sparse,
O(1)-spanner, both in Euclidean and hop distance. This
structure permits such routing which is optimal for
any geographic routing protocol. We also present a scheme
that performs load-balancing on the graph so that any
single back-bone is not overloaded by routing requests.

We consider the problem of maximizing the number of disjoint
dominating sets for certain classes of graphs that commonly
model wireless networks. We consider the problem on d-dimen.
unit ball graphs in Euclidean space, unit ball graphs
in doubling metric space, and growth independence graphs.
We almost (sadly, not quite)
give a constant-factor approximation to
the problem on all these graphs in O(1), O(log* n),
and O(log* n log Δ) rounds of distributed
computation, respectively.

We consider the problem of finding a TDMA slot assignment
for unicast communication in a wireless network on an
acyclic network. We give a randomized algorithm that computes
w.h.p., in poly-log(n) rounds of distributed computation, a
collision-free schedule of nearly 2*Δ time slots.

Non-rigid image registration (NIR) is an important tool
that allows for morphological comparisons in the presence
or anatomical variations. While many NIR algorithms have
been devised, they are difficult to evaluate in the
absence of point-wise inter-image correspondence.
We provide a framework that allows for evaluation of
these algorithms.

We present a composite data structure that is both
available and stabilizing. It is stabilizing in the
sense that it is resilient to transient faults occurring
in that over the course of time as the data structure
is operated upon, it corrects itself. It is available
in the sense that while it is possibly in an illegitimate
state, it is available to operations and a successful
operation is guaranteed to change the state of the
data structure in an expected manner.

Manuscripts/Reports:

Scheduling and Embedding Algorithms for Wireless
Networks

Ph.D. Thesis.

On k-domination of Graphs.

Ph.D. Comprehensive Examination.

A Self-Stabilizing Algorithm for a Weaker form
of Distributed Mutual Exclusion.